Data Packet Position Modulation System

ABSTRACT

A packet position modulation system includes a node configured to transmit a plurality of packets at corresponding time intervals. The node is configured to adjust, for at least one packet of the plurality of packets, the corresponding time interval to transmit the at least one packet. The system includes a base station configured to receive the plurality of packets from the node at corresponding time intervals, determine a difference between a previous time that a previous packet of the plurality of packets was received and a present time that a present packet of the plurality of packets was received, and recover coded data from the present packet based on the difference.

CROSS REFERENCE

This application claims the benefit of U.S. Provisional Application62/864,025, filed Jun. 20, 2019. The entire disclosure of the aboveapplication is incorporated herein by reference.

GOVERNMENT CLAUSE

This invention was made with government support under CNS1405273 awardedby the National Science Foundation. The government has certain rights inthe invention.

FIELD

The present disclosure relates to transfer of information using datapacket position modulation.

BACKGROUND

Transmit-only (Tx-only) networks have been proposed in recent years dueto their wide application for low-power communication in industrial,human health monitoring, and smart-home. Tx-only sensors cansignificantly reduce the cost of deployment since Tx-only sensors arecheaper compared to regular sensors. From an implementation perspective,the Tx-only sensor node consumes less energy due to the absence of thereceiving circuit.

This section provides background information related to the presentdisclosure which is not necessarily prior art.

SUMMARY

A packet position modulation system includes a node configured totransmit a plurality of packets at corresponding time intervals. Thenode is configured to adjust, for at least one packet of the pluralityof packets, the corresponding time interval to transmit the at least onepacket. The system includes a base station configured to receive theplurality of packets from the node at corresponding time intervals,determine a difference between a previous time that a previous packet ofthe plurality of packets was received and a present time that a presentpacket of the plurality of packets was received, and recover coded datafrom the present packet based on the difference.

In further aspects, the node includes a transmit time device configuredto receive sensor data from a sensor of the node, determine a delaybased on the sensor data, adjust the corresponding time interval basedon the determined delay, and transmit the at least one packet inaccordance with the adjusted time interval. In further aspects, thedifference indicates the sensor data.

In further aspects, the system includes an intermediary node including atransceiver and a sensor. In further aspects, the intermediary node isconfigured to receive the plurality of packets from the node and forwardthe plurality of packets to the base station. In further aspects, theintermediary node includes an intermediary sensor configured to sense anenvironment condition at a location of the intermediary sensor.

In further aspects, the node is configured to select a referenceinterval and, in response to an energy level exceeding a threshold by: apresent time equaling the reference interval less a next coded data,transmitting the present packet at the present time. In further aspects,the node is configured to, in response to the energy level being belowthe threshold by: the present time, transmitting the present packet at apostponed time, wherein the postposed time equals the reference intervalplus the next coded data.

In further aspects, the next coded data is a next data multiplied by apacket duration. In further aspects, the reference interval is selectedas greater than or equal to the corresponding time interval. In furtheraspects, the node includes a sensor and the plurality of packets includesensor data sensed by the sensor.

In further aspects, the base station includes a memory coupled to aprocessor. In further aspects, the memory stores instructions that, uponexecution, cause the processor to recover the coded data and store theprevious time that the previous packet of the plurality of packets wasreceived. In further aspects, the base station includes a displayconfigured to display the present packet and the recovered coded data.

A packet position modulation method includes transmitting, from a node,a plurality of packets at corresponding time intervals. The methodincludes adjusting, by the node, for a first packet of the plurality ofpackets, a first time interval that the first packet is transmitted andreceiving, at a base station, the first packet of the plurality ofpackets from the node at the first time interval. The method includesdetermining a difference between a previous time that a previous packetof the plurality of packets was received and a present time that thefirst packet of the plurality of packets was received and recoveringcoded data from the first packet based on the difference.

In further aspects, the method includes storing the present time thatthe first packet of the plurality of packets was received as theprevious time that the previous packet of the plurality of packets wasreceived. In further aspects, the method includes receiving, at thenode, sensor data from a sensor of the node, determining a time delaybased on the sensor data and a remaining energy level, adjusting thefirst time interval of the first packet based on the determined timedelay, and transmitting the first packet at the first time interval.

In further aspects, the difference indicates the sensor data. In furtheraspects, the method includes receiving, at an intermediary node, thefirst packet from the node and forwarding, from the intermediary node,the first packet to the base station. In further aspects, theintermediary node includes an intermediary sensor configured to sense anenvironment condition at a location of the intermediary sensor.

In further aspects, the method includes generating an intermediarypacket including the first packet and the environment condition sensedat the location of the intermediary sensor. In further aspects, theenvironment condition is included in the intermediary packet as thecoded data. In further aspects, the method includes forwarding, from theintermediary node, the intermediary packet to the base station.

A packet position modulation system includes a node configured totransmit sensor data at corresponding time intervals. The node includesa sensor and a transmit time device. The transmit time device isconfigured to receive sensor data from the sensor, determine a delaybased on the sensor data and a remaining energy level, adjust thecorresponding time interval based on the delay, and transmit the sensordata in accordance with the adjusted time interval. The system includesa base station configured to receive sensor data from the node at thecorresponding time interval. The base station includes a memory coupledto a processor. The memory stores instructions that, upon execution,cause the processor to determine an actual period between receivingprevious sensor data and the sensor data and calculate the sensor databased on the actual period.

DRAWINGS

The drawings described herein are for illustrative purposes only ofselected embodiments and not all possible implementations, and are notintended to limit the scope of the present disclosure.

FIG. 1 is a high-level network diagram of an example transmit-only(Tx-only) network.

FIG. 2A is an example Tx-only node.

FIG. 2B is an example cluster head.

FIG. 2C is an example base station.

FIG. 3 is a block diagram depicting transfer of data packets using anexample architecture of asymmetric packet position modulation (APPM).

FIG. 4A is a graphical depiction of example effective informationcapacity of APPM when using different packet lengths.

FIG. 4B is a graphical depiction of example effective informationcapacity of APPM when using different energy utilization rates.

FIG. 5 is a graphical depiction of an instantaneous and average energylevel of APPM and baseline.

FIG. 6 is a block diagram depicting transfer of data packets using anexample architecture of symmetric packet position modulation (SPPM).

FIG. 7A is a graphical depiction of example effective informationcapacity of SPPM when using different packet lengths.

FIG. 7B is a graphical depiction of example effective informationcapacity of SPPM when using different energy utilization rates.

FIG. 8 is a graphical depiction of an instantaneous and average energylevel of APPM, SPPM, and baseline.

FIG. 9 is a graphical depiction of a relative energy ratio of APPM andSPPM.

FIG. 10 is a graphical depiction of a relative delay ratio of APPM andSPPM.

FIG. 11 is a time chart depicting an example of an SPPM protocol overpre-allocated time-division multiple access (TDMA) slots (SPPM-PAD) or acollision-avoidance slotted protocol without overlap.

FIG. 12 is a time chart depicting an example of an SPPM protocol withimplicit slotting (SPPM-WIS) or with overlap.

FIG. 13 is a time chart depicting a definition for time shift distance.

FIGS. 14A-14B are graphical depictions showing changes in effectiveinformation capacity with an increasing time shift.

FIG. 15 is a graphical depiction showing changes in collisionprobability with an increasing time shift.

FIG. 16 is a graphical depiction showing a maximum Effective InformationTransfer Capacity (EITC) reduces with higher duty cycles for differentnetwork sizes.

FIG. 17 is a graphical depiction showing changes in effectiveinformation capacity with an increasing number of nodes in a network.

FIG. 18 is a graphical depiction showing changes in collisionprobability with an increasing number of nodes in a network.

FIG. 19 is a graphical depiction of a maximum attainable effectiveinformation transfer capacity (EITC) of SPPM-WIS as a result ofdifferent packet loss probabilities due to channel errors.

Corresponding reference numerals indicate corresponding parts throughoutthe several views of the drawings.

DETAILED DESCRIPTION

Example embodiments will now be described more fully with reference tothe accompanying drawings.

A packet position modulation (PPM) system or data PPM (DPPM) modulatesdata information in terms of inter-packet intervals without consumingany extra energy for Tx-only sensor networks and Internet of Things(IoTs) applications with thin energy budgets. The DPPM paradigm of thepresent disclosure is designed to enhance information capacity ofultra-low-bandwidth communication links used by energy-constrainedsensors and IoTs. Packet data transmissions in wireless sensor networksare often scheduled at regular intervals to reflect samplingrequirements, energy harvesting power availability, etc. In suchapplications, especially in energy-constrained scenarios, the time gapbetween consecutive packets is much larger (>100 times) than the packetduration. The core idea of the proposed DPPM is to leverage such largeinter-packet spacing for coding additional information without incurringany additional transmission energy expenses. Unlike prior capacityenhancement coding methods, the DPPM architectures improve theinformation transfer capacity of sensor networks without causing anyextra energy consumption and the complexity of hardware design.Analytical models and simulation results have been provided for theinformation transfer capacity improvement expectations.

Referring to FIG. 1, an example transmit-only (Tx-only) network isshown. A plurality of Tx-only nodes 104-1, 104-2, 104-3, 104-4, 104-5,and 104-6, collectively 104, are deployed massively to sense anenvironment and transmit packets to a base station 112. In variousimplementations, one or more cluster heads may act as intermediaries toforward sensed data from the Tx-only nodes 104 to the base station 112.Additionally, the cluster heads can also include environment sensors tocollect environment sensor data. The cluster heads are either wired orequipped with more complex, full-duplex radios. The cluster heads mayinclude a processor and associated memory for storing oranalyzing/organizing data received from the Tx-only nodes 104. Thecluster heads may also include a receive antenna for communicating withthe Tx-only nodes 104.

In various implementations, the Tx-only nodes 104 include an environmentsensor, for example, for sensing a temperature, a transmit antenna, anda small battery or simple energy harvesting device. Packet transmissionsin such a Tx-only sensor network are often scheduled for transmission inregular time periods or intervals to reflect sampling requirements orenergy harvesting power availability. In such applications, the amountof information transmitted is constrained by battery or harvestedenergy. The data rate is relatively low and the duration betweenconsecutive packets is much larger (>100 times) than the packet durationin order to maintain a long lifetime operation. It is always a challengeto enhance information capacity and throughput under limited energy.

Another major challenge faced in a Tx-only system is the chance ofpacket collision. Due to the difficulty of synchronization andcoordination between Tx-only sensor nodes, there is a high probabilityof collision in a multi-access environment. A high collision rate willhave a significant impact on the information transfer capacity of thesystem. It is therefore important to consider the chances of collisionfor maximizing the information transfer capacity.

The presently described packet position modulation (PPM) mechanism isbased on the modulation of inter-packet intervals for zero-energy datatransmission in Tx-only sensor networks and IoT applications driven withthin energy budget. Taking advantage of the inter-packet spacing toencode additional data information in terms of the time interval betweenconsecutive packets, PPM does not incur any extra energy expenses.

Wireless sensor networks (WSNs) have been successfully used to supportvarious applications. WSNs consist of small sensing devices that detectand respond to various types of input from the physical environment.Among the research towards improving information capacity in WSNs, asignificant disadvantage of multi-hop networks is the complexity ofsoftware protocols required to manage the network. Software protocolsmust manage the synchronization of sensing nodes, the discovery ofneighboring nodes, maintenance of multi-hop routes through the networkand fault tolerance for noisy node-to-node radio communication.Protocols for multi-hop environments must be highly fault-tolerant andable to re-transmit lost packets. Conversely, they must also maximizeenergy efficiency by avoiding unnecessary retransmissions of messages.As in the case of wireless networks, limited information capacity andbattery life are the main challenges. Among the research, theinformation transfer capacity has been improved significantly and theprotocols are designed accordingly for increasing the energy-efficiency.However, such research is based on the physical layer sensing capacityand multi-channel communication, which provides a high requirement forhardware design and implementation.

Tx-only sensors network, as one category of sensor networks, have beenproposed in recent years based on the fact that Tx-only sensors aresignificantly cheaper and simpler to build as well as being morereliable than a traditional multi-hop sensor network. It is concludedthat a Tx-only network can achieve equivalent performance to a networkwith transceiver nodes but at a much lower dollar cost for the hardwareand for lower power consumption. Moreover, Tx-only-based single-hopconfiguration includes no need for routing considerations and thereforesimpler protocol stack, lower delay, simpler time synchronization, andthe possibility of using centralized media access control.

DPPM architecture is designed based on the modulation of inter-packetsilence duration. An algorithm is deduced for maximum informationcapacity based on the consideration of multiple design parameters, suchas sensor density, energy utilization rate, hardware parameters, andpacket length, etc. Unlike prior coding methods for improvinginformation transfer capacity, the proposed DPPM method and thecorresponding MAC protocol described are differentiated in terms of zeroextra energy consumption and no need for receiving circuit on sensornodes, such as the Tx-only nodes 104 in FIG. 1.

Further, information capacity is improved when implementing DPPM bymaking the best use of limited battery or harvested energy. Themechanism can be implemented into application-specific low-power sensornetworks and IoT systems for impressive gains in the informationtransfer capacity, especially in sensor networks powered with slowenergy-harvesting sources.

Referring to FIG. 2A, an example Tx-only node 104 described in FIG. 1 isshown. The Tx-only node 104 includes a battery 204, a transmit antenna208, a sensor 212 for sensing an environmental condition, and a transmittime device 214. The transmit time device 214 is configured to convertdata information sensed by the sensor 212 into a time delay that iseither before or after the predetermined interval at which the datainformation is transmitted to the cluster head. In variousimplementations, the Tx-only node 104 transmits data information fromthe sensor 212 directly to the base station 112 at the predeterminedtime interval with the time delay corresponding to the data information.

In FIG. 2B, an example cluster head 116 is shown. In variousimplementations, the cluster head 116 may receive packets from theTx-only node 104 and forward the packet to the base station 112. Inother words, the cluster head 116 may act as an intermediary. Thecluster head 116 includes a battery 216, a transceiver 220, a sensor224, and a processor and associated memory 228. In variousimplementations, the cluster head 116 may exclude the processor andassociated memory 228 and simply forward sensor data receiver from thesensor 212 of the Tx-only node 104 to the base station 112 via thetransceiver 220. Referring to FIG. 2C, an example base station 112described in FIG. 1 is shown. The base station 112 includes a battery232, an Rx antenna 236, and a processor and associated memory 240. Invarious implementations, the batteries included in the Tx-only node 104,the cluster head 116, and/or the base station 112 may be energyharvesting devices.

The processor and memory 240 of the base station 112 is configured toidentify the time delay based on a difference between an expected timethe base station 112 will receive data information and when the basestation 112 does receive the data information. Based on the difference,the processor and memory 240 can determine the additional information ofthe sensor 212 being transmitted using the time delay. The base station112 may also include a display 244, through which a user can view theadditional information of the sensor 212 as well as view sensor data. Invarious implementations, the base station 112 is a computing device or amobile computing device that can store and collect sensor data forfurther analysis.

The DPPM paradigm is to enhance information transfer capacity ofcommunication links used by energy-constrained devices. Packettransmissions in low duty cycle networks are often scheduled astime-division multiple access (TDMA) slots, whose periodicity isdetermined based on application sampling requirements and the energyin-flow, often in the form of energy harvesting. The key idea of DPPM isto modulate the inter-packet spacing for coding additional informationwithout incurring additional transmission energy expenditures.

The DPPM based solution of the present disclosure is related tosingle-hop Tx-only networks in which a number of low-energy nodestransmit data to an aggregator. The architecture is first developed fora two-node point-to-point link, followed by a multipoint-to-pointmulti-access network. Detailed analytical and simulation models aredeveloped to demonstrate the performance of a symmetric and anasymmetric version DPPM. By carefully choosing the protocol parameters,DPPM can enhance the effective information transfer capacity of anultra-low duty cycle network by up to 65% in certain scenarios.

Additionally, low duty cycle networks have been extensively studied inthe sensor network literature for their ability to provideenergy-constrained data transport. Access control in such networks canbe TDMA or asynchronous non-TDMA based. While the asynchronousapproaches can operate in the absence of a centralized schedulingentity, they can suffer from energy wastage due to packet collisionswhich cannot be afforded in such energy-constrained networks. ATDMA-based approach, on the other hand, provides a collision-freesolution for medium access for low-cost embedded transceivers. For bothcases, the transmission duty cycle is very large when the energy inflowrate for a harvesting sensor is low.

Asymmetric Packet Position Modulation (APPM)

In a Tx-only network, the packet is normally transmitted at regular timeintervals T. The minimum time interval between the end of the previouspacket and the start of the present packet is normally hundreds of timesgreater than the packet duration. The sensor nodes go to sleeping modeduring the inter-packet intervals for energy-saving. In other words, thewide inter-packet intervals can be used for data modulation by adjustingthe transmission time of the next packets.

Therefore, the proposed DPPM system of the present disclosure works byshifting the position of a packet over time such that the amount ofshift encodes additional information to be sent. Consider a scenario inwhich a sensor node sends packets to a base station at a regularinterval T, which is the TDMA frame duration.

Now consider an example implementation in FIG. 3, in which instead oftransmitting packets at T intervals, each packet is postponed by τδ_(i)durations, where is the bit duration and δ_(i) (i=1, 2, . . . ) is theDPPM-coded data value. Since the base station expects a packet after Tduration since the last received packet, it interprets the shift τδ_(i)as the additional DPPM-coded information. With L-bit packets, thebaseline information transfer capacity is LIT bits per second. With DPPMas shown in FIG. 3, in addition to that baseline capacity, the node isable to send additional information values δ₁ and δ₂ by modulating thepositions of the second and the third packets respectively. Thisincreases the node's effective information transfer capacity (EITC) atno additional transmission energy costs. Formally stated, the capacityis enhanced from L bits in T duration to L+log₂ ^(δ) ^(i) bits inT+τδ_(i) duration.

Note that the bit durations in the inter-packet intervals can bedifferent from packet bit duration τ, and it should be chosen accordingto the accuracy of the clock within the sensor nodes. An accurate clockcan increase the number of bits between packets for higher informationcapacity. For simplicity, the bit duration in the inter-packet intervalsis also set as the same as packet bit duration τ for remainingdiscussion. The theoretical analysis can similarly be used for adifferent inter-packet bit length.

In APPM, it is assumed that the maximum delayed bit duration is Δ (inbit durations), which can represent the data within the range [0, Δ−1].Each data value i (i∈[0, Δ−1]) is modulated as the delayed (i+1) bitdurations. The data value is uniformly distributed within [0, Δ−1]. Forother distributions, the information transfer capacity of APPM can bededuced similarly according to the following procedures. The averagetime duration between the start bit of the previous packet and the startbit of the present packet is:

$\begin{matrix}{T_{avg} = {{T + {\frac{1 + \Delta}{2} \cdot \tau}} = {\frac{L \cdot E}{W} + {\frac{1 + \Delta}{2} \cdot {\tau.}}}}} & (1)\end{matrix}$

The average encoded data information per packet in terms of bit can bedivided into two parts, one is L bits packet itself, and the other partis the extra bit information with APPM. The average encoded datainformation per packet is:

$\begin{matrix}{{D_{avg} = {{L + \frac{{\log_{2}1} + {\log_{2}2\text{...}} + {\log_{2}\Delta}}{\Delta}} = {L + \frac{\log_{2}\left( {\Delta!} \right)}{\Delta}}}}.} & (2)\end{matrix}$

The information transfer capacity can be deduced from Eq. (1) and Eq.(2) as:

$\begin{matrix}{C_{APPM} = {\left( {L + \frac{\log_{2}\left( {\Delta!} \right)}{\Delta}} \right)/{\left( {\frac{L \cdot E}{W} + {\frac{1 + \Delta}{2} \cdot \tau}} \right).}}} & (3)\end{matrix}$

The ratio of C_(APPM) to the baseline information capacity C_(BL) isdefined as Effective Channel Capacity (ECC) η=C_(APPM)/C_(BL). Referringto FIG. 4A, ECC with both the theoretical results and the simulationresults under the different length of the packet with a single Tx-onlynode/sensor is shown. For an energy utilization rate W=0.1 mW, theinformation transfer capacity of APPM can achieve 20% to 40% improvementcompared to the baseline information capacity when L is small, forexample, L=16, 32.

FIG. 4B shows the simulated and the theoretical ECC with packet lengthL=32 bits under different energy utilization rates. Note that themaximum inter-packet interval used in the simulation is T=5 seconds,which is a relatively small inter-packet interval compared with the realapplication. A relatively smaller utilization rate leads to a largertime interval T between packets, which APPM can take advantage of forachieving a higher information transfer capacity.

However, APPM postpones each packet for improving information capacity.The average delay for each packet is

${t_{APPM} = {\frac{1 + \Delta}{2} \cdot \tau}},$

which cannot be avoided in APPM process. Simultaneously, the latency ofdata transmission saves the extra cumulative energy which is not usedfor improving information capacity. FIG. 5 shows the instantaneousenergy level and the average energy level within a sensor node with L=16bits and W=0.2 mW. According to the simulation results in FIG. 5, theaverage energy level keeps increasing with each packet transmission.

Symmetric Packet Position Modulation (SPPM)

A new coding scheme, SPPM, is based on APPM that further improvesinformation transfer capacity. Assume that inter-packet interval T′ ispredefined by a sensor network, such as the network described withrespect to FIG. 1. The value of T′ can be determined by the applicationrequirement and energy utilization rate. The minimum value of T′ is thebaseline inter-packet interval T, namely, T′≥T=L·E/W. SPPM is designedas shown in FIG. 6.

A Tx-only node depends on the energy utilization rate to arrange theschedule for the next packet transmission. For example, as shown anddescribed in FIG. 2A, the transmit time device 214 included in theTx-only node 104 is configured to determine the packet transmissionschedule and adjusts transmissions to correspond to the data that isbeing transmitted. Then, the processor and memory 228 of the basestation 112 converts the adjusted transmission schedule back into thedata to which it corresponds. In this way, the Tx-only node 104 canconvert data into a schedule and the base station 112 can convert thetransmission schedule deviation back into the data from the Tx-only node104.

If the energy can be cumulated sufficiently from the present until thetime T′−τδ_(i) (where δ_(i) is an arbitrary data vale) for sending apacket, the transmitter will send the packet at the time T′−τδ_(i).Otherwise, the next packet will be sent at the time T′+τδ_(i). Thereceivers can modulate the data information from the predefinedreference time interval T′. If the packet is received before the end oftime duration T′, the packet has been preponed (brought forward).Otherwise, the packet has been postponed.

FIG. 6 shows an example of SPPM architecture. For data value (1, thesensor node does not cumulate enough energy for preponing the packet.Thus, the packet is postponed for time duration τδ₁. But for the secondpacket transmission, enough energy has been stored for sending thepacket earlier. The second packet is preponed for time duration τδ₂before the end of time T′.

In each packet transmission, energy can be efficiently used in SPPM forscheduling and transmitting the packet. Simultaneously, delay can bealleviated from the early transmission (prepone) of the packet. Assumethat Δ is the time shift bit duration for SPPM, and Δ≤T′/τ. An arbitrarydata value δ_(i) follows uniform distribution within the range of [0,Δ−1]. There is an optimal T′ for maximizing information capacity. Thecalculation of the optimal T′ can be realized as follows that, whenT′≤T, since the baseline inter-packet interval T is determined by theenergy utilization rate W and hardware setup, namely that T=L·E/W. InSPPM, a sensor node can store the extra energy when after postponing apacket. The stored energy can be used for preponing of next packet. Eventhough T′≤T, the overall average inter-packet interval is still T,because T is, on average, the minimum inter-packet interval based onenergy utilization. A sensor node has more than 50% probability topostpone the packet and less than 50% probability to prepone the packet.The average encoded data information per packet is

${D_{avg} = {L + \frac{\log_{2}\left( {\Delta!} \right)}{\Delta}}},$

which is the same as the one in APPM. Therefore, the informationtransfer capacity of SPPM is

$\begin{matrix}{C_{SPPM} = {\left( {L + \frac{\log_{2}\left( \Delta^{1} \right)}{\Delta}} \right)/{\frac{L \cdot E}{W}.}}} & (4)\end{matrix}$

Then, when

${T < {T\prime} \leq {\left( {\frac{2{L \cdot E}}{W \cdot \tau} - {2L} + 1} \right)\tau}},$

if the reference interval T′ in SPPM is greater than T, the cumulatedenergy after T′ time period is greater than the required energy forsending a packet. Thus, a sensor node can be more likely to prepone apacket than postpone a packet. In the extreme case, T′ is large enoughthat all the packets can be sent in the preponing mode. When

${T^{\prime} = {\left( {\frac{2{L \cdot E}}{W \cdot \tau} - {2L} + 1} \right)\tau}},$

after all the packet has been preponed, the average inter-packetinterval is equal to baseline T. However, the average inter-packetinterval is constrained by the energy utilization rate, and it cannot beless than T. Therefore, the average inter-packet interval T_(avg) isstill T and the information transfer capacity is equal to:

$\begin{matrix}{{C_{SPPM} = {\left( {L + \frac{\log_{2}\left( {\Delta!} \right)}{\Delta}} \right)/T}}.} & (5)\end{matrix}$

When

${T^{\prime} = {{\left( {\frac{2{L \cdot E}}{W \cdot \tau} - {2L} + 1} \right)\tau \mspace{14mu} {and}\mspace{14mu} \Delta} = {\frac{2{L \cdot E}}{W \cdot \tau} - {2L} + 1}}},$

the information transfer capacity achieves the maximum value with themaximum data transmission load in T time period.

Further, when

${T^{\prime} > {\left( {\frac{2{L \cdot E}}{W \cdot \tau} - {2L} + 1} \right)\tau}},$

since the reference interval T′ in SPPM is large enough that all thedata can be preponed, the average inter-packet interval is

$T_{avg} = {T^{\prime} - {\left( {\frac{\frac{T^{\prime}}{\tau} + 1}{2} - L} \right).}}$

Therefore, the inter-packet interval is greater than T. The informationtransfer capacity in this case is:

$\begin{matrix}{C_{SPPM} = {\left( {L + \frac{\log_{2}\left( {\Delta!} \right)}{\Delta}} \right)/\left( {T^{\prime} - \left( {\frac{\frac{T^{\prime}}{\tau} + 1}{2} - L} \right)} \right)}} & (6)\end{matrix}$

The amount of modulated information is increasing because the increaseof T′ leads to the increase of

${\Delta \left( {\Delta = {\frac{T}{\tau} - L}} \right)},$

which means a wider space for modulation. But, the average inter-packetinterval is increasing faster than the amount of modulated information.The overall process incurs the decreasing of the information transfercapacity.

FIG. 7A shows ECC

$\left( {{defined}\mspace{14mu} {as}\mspace{14mu} {\eta = \frac{c_{SPPM}}{c_{BL}}}} \right)$

with SPPM mechanism under energy utilization rate W=0.1 mW for differentlength of packet. The simulation results agree well with the theoreticalanalysis. When

${{\Delta = {\frac{2{L \cdot E}}{W \cdot \tau} - {2L} +}}1},$

the information capacity obtains the maximum value.

ECC using SPPM under packet length L=32 bits with different energyutilization rates is shown in FIG. 7B. It shows that the relativelysmaller energy utilization rate can benefit the performance of SPPMinformation capacity. Compared with APPM, SPPM can achieve more than 15%improvement on information capacity.

FIG. 8 shows the instantaneous energy level and the average energy levelwithin a sensor node with L=16 bits and W=0.2 mW. The average energylevel of SPPM oscillates along the baseline curve. It can make the mostof energy for scheduling packet transmission.

For further comparing the performance of SPPM with APPM, the relativeenergy ratio (R_(E)) is used to show the energy utilization, which isdefined as the ratio of the remaining energy after each packettransmission (E_(i)) to the product of the packet sequence number (i)and each packet energy consumption (E_(packet)) in Eq. (7). Similarly,the relative delay ratio (R_(t)) is defined as the ratio of the timeafter sending each packet to the product of the packet sequence number(i) and the baseline inter-packet interval (T) as shown in Eq. (7).

$\begin{matrix}\left\{ \begin{matrix}{R_{E} = \frac{E_{i}}{i \cdot E_{packet}}} \\{R_{t} = \frac{t_{i}}{i \cdot T}}\end{matrix} \right. & (7)\end{matrix}$

FIG. 9 shows the relative energy ratio of APPM and SPPM with the sametime shift Δ=10208 bit durations

$\left( {\Delta = {\Delta_{T} = {\frac{L \cdot E}{W \cdot \tau} - L}}} \right)$

under L=32 bits and W=0.1 mW. It can be seen that the relative energyratio of SPPM converges to zero. Because after each packet transmissionin SPPM, there is not enough energy left for immediately sending thenext packet. On the contrary, the relative energy ratio of APPMconverges to 0.5 because APPM retains the amount of energy

$\frac{\Delta + 1}{2}W$

compared to the baseline after each packet transmission on average andthis amount of energy is not used for future packet transmission. Inother words, SPPM can make the best use of energy for reasonablyscheduling the packet transmission.

FIG. 10 shows the relative delay ratio of APPM and SPPM with the sametime shift parameter Δ=A_(T). Since APPM incurs the time delay

$\frac{\Delta + 1}{2}\tau$

after each packet transmission, the delay ratio converges to the averagedelay after a certain number of packet transmission. The delay ratio ofSPPM converges to zero due to the preponing transmitting of packets,which compensates the delay from postponing the packets. In variousimplementations, SPPM may achieve better performance on informationcapacity, energy utilization, and transmission delay than APPM.

Mac Layer Protocol Design for SPPM

The performance of SPPM is analyzed with a single transmitter so far. Invarious implementations, MAC layer protocols may be implemented for SPPMto enhance the information transfer capacity in a Tx-only network.Assume that N number of nodes in a Tx-only network. In order to avoidthe collision of packets from different nodes, the inter-packet intervalframe T (T=L·E/W) is divided into N number of slots. Each slot isassigned for each node. The packet from a specific node can only appearin the slot which is assigned for that node.

FIG. 11 describes a schematic of an SPPM protocol over pre-allocatedTDMA slots (SPPM-PAD) or a collision-avoidance SPPM without overlapping(CASPPM). For example, there are three nodes in a Tx-only network. Eachnode is assigned a unique slot for its own packet transmission. Themiddle point is reference point for SPPM purpose. The data isrepresented as the time interval between the received packet and thereference point. It is not possible that two packets appear in the sameslot. Such a protocol can effectively avoid collision of packets fromdifferent nodes. The maximum time shifts that a node can use for SPPM is

$\left\lfloor \left( {\frac{T}{N \cdot \tau} - L} \right) \right\rfloor$

in bit durations. The information transfer capacity per node is:

$\begin{matrix}{C_{\underset{SPPM}{TDMA}} = {\left( {L + \frac{\log_{2}\left( {\left\lfloor {\left( {\frac{T}{N \cdot \tau} - L} \right)/2} \right\rfloor!} \right)}{\Delta}} \right)/\frac{L \cdot E}{W}}} & (8)\end{matrix}$

SPPM-PAD protocol increases the information transfer capacity byavoiding the expense of packet collisions. However, such a protocolcannot support a network with a large number of nodes, whichdramatically decreases the space for SPPM and information capacity ofthe network. Moreover, SPPM-PAD necessitates a high synchronizationrequirement for the nodes within the network. The protocol assumes thatall the nodes are synchronized to the bit level. In order to eliminatesynchronization requirement and further enhance the information transfercapacity, a new MAC protocol is described below.

In a Tx-only network, the initial positions of the nodes are randomlydistributed in a T′ (T′≥T) time period. The nodes in the network areindependent from each other. After a node sends a packet, the nextpacket is scheduled according its present packet location, eitherpreponing or postponing the next packet.

The primary limitation of SPPM-PAD is that the amount of allowed shiftis bounded to only half the slot duration in each direction, whichlimits the maximum possible information transfer capacity, especially atsmall node population. Also, the mechanism requires all nodes to betightly absolute time-synchronized among each other and the basestation. This is particularly challenging due to: a) high clock driftsin inexpensive embedded nodes, and b) lack of reception ability of theTx-only nodes by periodically synchronizing with the base station. Thefollowing protocol addresses these.

FIG. 12 shows an example of SPPM protocol with implicit slotting(SPPM-WIS). There is no explicit slot allocated to a node in thisversion. Rather, once a node picks a transmission time, it keeps sendingpackets periodically with the baseline interval

${T = {L \cdot \frac{E}{W}}},$

which depends on energy harvesting rate, packet length, and transmissionenergy budget. With this strategy, the nodes do not have to be absolutetime synchronized with other nodes. It is sufficient to self-synchronizein a relative sense so that the receiver is able to measure anytransmission time shift in order to decode the additional informationcoded by SPPM-WIS.

A node is allowed to shift (i.e., left or right depending on the energyavailability) a transmission with respect to its last transmission timeby up to the reference duration T′ as defined and dimensioned above.Additionally, the preponing and postponing of transmissions based onavailable energy is performed the same way as presented above. Anexample operation of SPPM-WIS is shown in FIG. 12.

Unlike in SPPM-PAD, there can be collisions in SPPM-WIS. However, sincethe range of encoded data value is

$\left( {\frac{T^{\prime}}{\tau} - L} \right),$

which is larger than that in SPPM-PAD, this version can achieve a higherinformation transfer capacity, especially at smaller node populations.This advantage diminishes due to frequent collisions in larger networks.A detailed analytical model for the collision probability andperformance of SPPM-WIS, along with an algorithm for choosing theoptimal time shifts for the maximum information transfer capacity, aredescribed later.

Since each node can use the whole frame to do packet positionmodulation, the range of encoded data values is enhanced compared withthe previous MAC protocol. The sensor nodes in a Tx-only network canmake the best use of time shifts duration between packets to achieve themaximum information capacity. However, such a strategy incurs thecollision between the packets from different sensor nodes, which canreduce the information transfer capacity. An algorithm is developed toobtain the maximum information transfer capacity with a specific groupof parameters (energy consumption per bit, packet size number of nodes,and energy utilization rate).

In order to obtain the maximum information transfer capacity, thecollision probability for a specific group of parameters is analyzed.First, a transition matrix is used to describe the position of packetsfor the later analysis of collision probability. Time shift

$\Delta \left( {\Delta \leq \frac{T^{\prime}}{\tau}} \right)$

is used to encode the data value during SPPM. The time shift distance Dis defined as the bit duration distance from the start of the presentlymodulated packet to the start of the corresponding baseline packet withthe same sequence number as shown in FIG. 13.

FIG. 13 shows the schematic of the definition for time shift distance D.In FIG. 13, a sensor node does not have enough energy to prepone thefirst packet at the time T′−τδ₁, it postpones sending the first packetuntil the time t₁=T′+τβ₁. The distance for the first packet isD=(t₁−T)/τ+L in bit durations and the energy level is D₁τW. Similarly,the distances for the second and the third packets are D₂=(t₂−2T)/τ+Land D₂=(t₃−3T)/τ+L bit durations, respectively, and the correspondingenergy levels are D₂τW and D₃τW, respectively.

The general expression for the distance of the packet isD_(i)=(t_(i)−iT)/τ+L. Due to the definition for baseline packettransmission, the baseline inter-packet interval T is the minimum timeinterval between two packets on average. Therefore, D_(i) is alwayslarger or equal to zero. The maximum distance D_(i)=Δ+Δ−1=2Δ−1, whichcan be obtained from the fact that the three consecutive distances areD_(i−2)=0, D_(i−1)=Δ−1 with cumulative energy level (Δ−1)τW (postponingΔ−1 bit durations), and D_(i)=2Δ−1 with cumulative energy level (2Δ−1)τW(postponing Δ bit durations), respectively. The distance D defines 2Δstates (from 0 to 2Δ−1) in a state machine, which represents the packetposition during SPPM, and the transition between the states depends onthe available energy level and time shift parameter Δ.

Take

$\Delta = \frac{T}{\tau}$

for instance, namely T′=T. When the distance of a packet is D=0 (thepresent state is 0), the probability vector that the packet transitsfrom state 0 to all the other states (0, 1, . . . , 2Δ−1) where the nextpacket transmission can appear is

$\begin{matrix}\left\lbrack {0,\overset{\Delta}{\overset{}{\frac{1}{\Delta},\frac{1}{\Delta},\ldots \mspace{14mu},\frac{1}{\Delta}}},0,0,\ldots \mspace{14mu},0} \right\rbrack & (9)\end{matrix}$

where the vector in Eq. (9) means a the state can transit to the state1, 2, . . . , A with the probability

$\frac{1}{\Delta}.$

Similarly, when the present packet position is D=1, the probabilityvector that the packet position transits from state 1 to all the otherstates (0, 1, . . . , 2Δ−1) where the next packet transmission canappear is:

$\begin{matrix}\left\lbrack {\overset{\Delta + 2}{\overset{}{\frac{1}{\Delta},0,0,\frac{1}{\Delta},\ldots \mspace{14mu},\frac{1}{\Delta}}},0,0,\ldots \mspace{14mu},0} \right\rbrack & (10)\end{matrix}$

where the above transition vector is obtained from the analysis ofcumulated energy.

Since the present state is D=1, which can be considered as the packet ispostponed for one bit duration from state D=0, and thus the presentenergy level at D=1 is E_(D=1)=1·τ·W=τW. Based on the PPM rules, thenext state of packet position cannot be the state 2, because if theenergy level is enough, the prepone of the packet is preferred insteadof postponing. Simultaneously, about the next states of the state 1, itcannot be itself due to the definition of PPM. Therefore, the vector inEq. (10) is obtained about the transition probability of the state 1 toall the other states where the next packet transmission can appear.

When the state of packet position D=i (0≤i≤2Δ−1) and 2i<2Δ−1, theprobability that the state transits from D=i to D=i+1, i+2, . . . , 2iis equal to zero because the packet is preponed if energy level isavailable. When a data value δ (δ>i) is modulated with a packet, theprobability that the state transits from D=i to D=2i+1, 2i+2, . . . ,Δ+i is

$\frac{1}{\Delta}$

because the harvested energy is not enough to support preponing thepacket with the value δ>i and the postpone is necessary for modulating δwith packet. Since the maximum value of modulated data is δ≤Δ−1, thestate D=i has zero probability to convert itself to the state D=Δ+i+1,Δ+i+2, . . . , 2Δ−1. In conclusion, the probability vector that thepacket position transits from state i (2i<2Δ−1) to all the other states(0, 1, . . . , 2Δ−1) for the next packet transmission is shown in Eq.(11):

$\begin{matrix}\left\lbrack {\overset{i - 1}{\overset{}{\frac{1}{\Delta},\frac{1}{\Delta},\ldots \mspace{14mu},\frac{1}{\Delta}}},\overset{i + 1}{\overset{}{0,0,\ldots \mspace{14mu},0}},\overset{\Delta - i}{\overset{}{\frac{1}{\Delta},\frac{1}{\Delta},\ldots \mspace{14mu},\frac{1}{\Delta}}},\overset{\Delta - i - 1}{\overset{}{0,0,\ldots \mspace{14mu},0}}} \right\rbrack & (11)\end{matrix}$

When the state of packet position D=i (0≤i≤2Δ−1) and 2i≥2Δ−1, theprobability vector that the packet position transits from state i to allthe other states (0, 1, . . . , 2Δ−1) for the next packet transmissionis shown in Eq. (12):

$\begin{matrix}\left\lbrack {\overset{i - \Delta}{\overset{}{0,0,\ldots \mspace{14mu},0}},\overset{\Delta}{\overset{}{\frac{1}{\Delta},\frac{1}{\Delta},\ldots \mspace{11mu},\frac{1}{\Delta}}},\overset{{2\Delta} - i - 1}{\overset{}{0,0,\ldots \mspace{14mu},0}}} \right\rbrack & (12)\end{matrix}$

A comprehensive one-step transmission matrix between 2Δ−1 states can bedrawn based on the above analysis as:

$\begin{matrix}{\mspace{115mu} {{{0\mspace{59mu} 1\mspace{56mu} 2\mspace{56mu} 3\mspace{40mu} \ldots \mspace{25mu} \Delta} - {1\mspace{31mu} \Delta \mspace{31mu} \Delta} + {1\mspace{14mu} \Delta} + {2\mspace{20mu} \ldots \mspace{11mu} 2\Delta} - {2\mspace{14mu} 2\Delta} - 1}{{\begin{matrix}0 \\1 \\\vdots \\{\Delta - 1} \\\Delta \\{\Delta + 1} \\\vdots \\{{2\Delta} - 1}\end{matrix}\begin{bmatrix}0 & {1/\Delta} & {1/\Delta} & {1/\Delta} & \ldots & {1/\Delta} & {1/\Delta} & 0 & 0 & \ldots & 0 & 0 \\{1/\Delta} & 0 & 0 & {1/\Delta} & \ldots & {1/\Delta} & {1/\Delta} & {1/\Delta} & 0 & \ldots & 0 & 0 \\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\{1/\Delta} & {1/\Delta} & {1/\Delta} & {1/\Delta} & \ldots & 0 & 0 & 0 & 0 & \ldots & 0 & {1/\Delta} \\{1/\Delta} & {1/\Delta} & {1/\Delta} & {1/\Delta} & \ldots & {1/\Delta} & 0 & 0 & 0 & \ldots & 0 & 0 \\0 & {1/\Delta} & {1/\Delta} & {1/\Delta} & \ldots & {1/\Delta} & {1/\Delta} & 0 & 0 & \ldots & 0 & 0 \\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\0 & 0 & 0 & 0 & \ldots & {1/\Delta} & {1/\Delta} & {1/\Delta} & {1/\Delta} & \ldots & {1/\Delta} & 0\end{bmatrix}} = P}}} & (13)\end{matrix}$

where the transition matrix P in Eq. (13) shows one-step transitionprobability of Markov chain [20] between 2Δ states with Σ_(j=1)^(2Δ)p_(ij)=1.

For any two states a and b (a, b∈{D₀, D₁, . . . , D_(2Δ−1),}), it ispossible to transfer to any state from any other state (p_(ij)≥0).Therefore, the corresponding Markov Chain {x_(n)} (n=0, 1, . . . ,x_(n)∈{D₀, D₁, . . . , D_(2Δ−1)}) of the matrix P is irreducible. Astate a has period k=1 if any return to state a must occur in multiplesof k time steps, the state is said to be aperiodic. For example,consider starting at a state D₂ travelling along the arrows, and endingback at D₂ in 2 steps (D₂→D₁→D₂), or 3 steps (D₂→D₄→D₁→D₂), so state 2is aperiodic. If An irreducible Markov chain only needs one aperiodicstate to imply all states are aperiodic. If a Markov chain isirreducible and aperiodic, it has a limiting probability which is theunique solution of π=πP. π is the equilibrium distribution of the chain,and it is also the steady-state distribution of packet position from aTx-only sensor node. If the initial position of nodes is known in thetimeline, the packet positions of each node can be drawn through theabove analysis, and the probability for packet position at each statecan be determined by calculating the equilibrium distribution π.

The equilibrium distribution π describes the theoretical analysis ofpacket position distribution of two nodes, which completely agrees withthe simulation results. The overlapping area of PMF between two nodes'packets are collision area, which is the big impact on the performanceof information transfer capacity for a Tx-only network.

As described above, it is known that

$T^{\prime} = {\left( {\frac{2{L \cdot E}}{W \cdot \tau} - {2L} + 1} \right)\tau}$

is corresponding to the maximum information transfer capacity with asingle Tx-only sensor, and the average inter-packet interval is equal toT. Since packets from all the nodes are randomly distributed in a packetframe T, collision cannot be avoided. The collision probability can beanalyzed according to the overlapping areas of packet position fromdifferent transmitters.

Since the packet position distribution can be obtained through thetransition matrix in Eq. (13), the steady-state for the Markov chain wcan be calculated by getting non-zero solution vector of the linearhomogeneous system of equations π=πP. The solution vector is theequilibrium distribution of packet position and can be easily checked byinserting the solution π in the equation to see if w satisfies theequation π=πP.

Sensor nodes are randomly distributed in a packet frame T. To calculatethe collision probability for a node, assume that time shift parameter Δ(in bit durations) is used for each node during SPPM, the maximum valuecan be achieved when

$\Delta \leq {\frac{2{L \cdot E}}{W \cdot \tau} - {2L} + 1.}$

N number of nodes with the node Ids∈[1,N] are randomly distributedwithin the packet frame T, the states (packet position) for each node isdefined as [D₀ ⁽¹⁾, D₁ ⁽¹⁾, . . . , D_(2Δ−1) ⁽¹⁾] for Node-1, the states

$\left\lbrack {D_{\frac{T}{\tau \; N}}^{(2)},D_{\frac{T}{\tau \; N} + 1}^{(2)},\ldots \mspace{14mu},D_{\frac{T}{\tau \; N} + {2\Delta} - 1}^{(2)}} \right\rbrack$

for Node-2, and so on, the states

$\left\lbrack {D_{\frac{T{({N - 1})}}{\tau \; N}}^{(N)},D_{\frac{T{({N - 1})}}{\tau \; N} + 1}^{(N)},\ldots \mspace{14mu},D_{\frac{T{({N - 1})}}{\tau \; N} + {2\Delta} - 1}^{(N)}} \right\rbrack$

for Node-N. For example, when N=2 and Δ=240 bit durations, the statesrange for all the nodes is from 0 to

${\frac{T\left( {N - 1} \right)}{\tau \; N} + {2\Delta} - 1} = {599\mspace{14mu} {bit}\mspace{14mu} {{durations}.}}$

The probability of packet position of each node across the overallstates for all the nodes is defined as:

$\begin{matrix}\left\{ \begin{matrix}\left\lbrack {P_{D_{0}}^{(1)},P_{D_{1}}^{(1)},\ \ldots \mspace{14mu},P_{D_{{2\Delta} - 1}}^{(1)},{\ldots \mspace{20mu} P_{D_{\frac{T{({N - 1})}}{\tau N} + {2\Delta} - 1}}^{(1)}}} \right\rbrack \\\left\lbrack {P_{D_{0}}^{(2)},P_{D_{1}}^{(2)},\ldots \mspace{14mu},P_{D_{\frac{T}{\tau N}}}^{(2)},{\ldots \mspace{20mu} P_{D_{\frac{T{({N - 1})}}{\tau N} + {2\Delta} - 1}}^{(2)}}} \right\rbrack \\\vdots \\\left\lbrack {P_{D_{0}}^{(N)},P_{D_{1}}^{(N)},\ \ldots \mspace{14mu},P_{D_{{2\Delta} - 1}}^{(N)},{\ldots \mspace{14mu} P_{D_{\frac{T{({N - 1})}}{\tau N} + {2\Delta} - 1}}^{(N)}}} \right\rbrack\end{matrix} \right. & (14)\end{matrix}$

The probability of each node's packet at all the possible packetpositions in the whole network has been listed above for the purpose ofcollision probability calculation. Starting the analysis from the signalnode and take Node-1, for example. First, calculate the collisionprobability of two nodes (Node-1 and Node-2). Assume that when thepacket from Node-1 appears at state D_(i), collision can only happenwhen the packet from Node-2 appear at states between D_(i−L+1) andD_(i+L−1). Since collision between two nodes are considered, all theother nodes, except Node-1 and Node-2, should not appear betweenD_(i−L+1) and D_(i+L−1). Therefore, the collision probability betweenNode-1 and Node-2 is expressed as:

$\begin{matrix}{{P_{D_{i}}^{(1)}\left( {\sum_{D_{j} \in {\lbrack{D_{i - L + 1},D_{i + L - 1}}\rbrack}}P_{D_{j}}^{(2)}} \right)}\left( {\prod_{a \in {{\lbrack{1,N}\rbrack} \smallsetminus {\{{1,2}\}}}}\left( {\sum_{D_{k} \in {{\lbrack{0,{\frac{T{({N - 1})}}{\tau N} + {2\Delta} - 1}}\rbrack} \smallsetminus {\lbrack{D_{i - L + 1},D_{i + L - 1}}\rbrack}}}P_{D_{k}}^{(a)}} \right)} \right)} & (15)\end{matrix}$

where a∈[1, N]\{1,2} denotes a∈[1, N] and a∉{1,2}, the complement of{1,2} in [1, N]. It is the same meaning for the expression

$D_{k} \in {\left\lbrack {0,{\frac{T\left( {N - 1} \right)}{\tau N} + {2\Delta} - 1}} \right\rbrack \smallsetminus {\left\lbrack {D_{i - L + 1},D_{i + L - 1}} \right\rbrack.}}$

Since the collision probability of the packets between Node-1 and Node-2is calculated in Eq. (15), the collision probability of the packetsbetween Node-1 and one of any other nodes can also be calculatedaccordingly as:

$\begin{matrix}{{P_{D_{i}}^{(1)}\left( {\sum_{D_{j} \in {\lbrack{D_{i - L + 1},D_{i + L - 1}}\rbrack}}P_{D_{j}}^{(b)}} \right)}\left( {\prod_{a \in {{\lbrack{1,N}\rbrack} \smallsetminus {\{{1,b}\}}}}\left( {\sum_{D_{k} \in {{\lbrack{0,{\frac{T{({N - 1})}}{\tau N} + {2\Delta} - 1}}\rbrack} \smallsetminus {\lbrack{D_{i - L + 1},D_{i + L - 1}}\rbrack}}}P_{D_{k}}^{(a)}} \right)} \right)} & (16)\end{matrix}$

The above calculates in Eq. (16) shows the collision probability ofNode-1 with one of any other nodes when the packets from Node-1 appearat state D₀. Similarly, the collision probability of Node-1 with one ofany other nodes when the packets from Node-1 appear at all the states

$D_{0},D_{1},{\ldots \mspace{14mu} D_{\frac{T{({N - 1})}}{\tau N} + {2\Delta} - 1}}$

can be expressed as:

$\begin{matrix}{P_{2}^{(1)} = {\sum\limits_{D_{i} = D_{0}}^{\frac{D_{T{({N - 1})}}}{\tau \; N} + {2\Delta} - 1}{{P_{D_{i}}^{(1)}\left( {\sum\limits_{\underset{b \neq 1}{D_{j} \in {\lbrack{D_{i - L + 1},D_{i + L - 1}}\rbrack}}}P_{D_{j}}^{(b)}} \right)}\left( {\prod\limits_{a \in {{\lbrack{1,N}\rbrack}\backslash {\{{1,b}\}}}}\; \left( {\sum\limits_{D_{k} \in {{\lbrack{0,{\frac{T{({N - 1})}}{\tau \; N} + {2\Delta} - 1}}\rbrack} \smallsetminus {\lbrack{D_{i - L + 1},D_{i + L - 1}}\rbrack}}}P_{D_{k}}^{(\alpha)}} \right)} \right)}}} & (17)\end{matrix}$

Second, the probability P₃ ⁽¹⁾ of the packets from Node-1 collided withthe packets from two of any other nodes together can be according to theabove analysis and be expressed as:

$\begin{matrix}{P_{3}^{(1)} = {\sum_{D_{i} = D_{0}}^{\frac{D_{T{({N - 1})}}}{\tau \; N} + {2\Delta} - 1}{P_{D_{i}}^{(1)}{\quad{\left( {\sum_{\underset{b \neq 1}{D_{j} \in {\lbrack{D_{i - L + 1},D_{i + L - 1}}\rbrack}}}{P_{D_{j}}^{(b)}{\sum_{\underset{c \neq 1}{D_{j} \in {\lbrack{D_{i - L + 1},D_{i + L - 1}}\rbrack}}}P_{D_{j}}^{(c)}}}} \right)*\left( {\prod_{a \in {{\lbrack{1,N}\rbrack}\backslash {\{{1,b,c}\}}}}\; \left( {\sum_{D_{k} \in {{\lbrack{0,{\frac{T{({N - 1})}}{\tau \; N} + {2\Delta} - 1}}\rbrack}{{\backslash\lbrack}{D_{i - L + 1},D_{i + L - 1}}\rbrack}}}P_{D_{k}}^{(a)}} \right)} \right)}}}}} & (18)\end{matrix}$

Third, the probability of the packets from Node-1 collided with thepackets from 3, 4, . . . , (N−1) of any other nodes together can bededuced accordingly as P₄ ⁽¹⁾, P₅ ⁽¹⁾, . . . , P_(N) ⁽¹⁾. Finally, thecollision matrix of all the nodes can be expressed as:

$\begin{matrix}\left\{ \begin{matrix}\left\lbrack {P_{2}^{(1)},P_{3}^{(1)},\ldots \mspace{14mu},P_{N}^{(1)}} \right\rbrack \\\left\lbrack {P_{2}^{(2)},P_{2}^{(2)},\ldots \mspace{14mu},P_{N}^{(2)}} \right\rbrack \\\vdots \\\left\lbrack {P_{2}^{(N)},P_{3}^{(N)},\ldots \mspace{14mu},P_{N}^{(N)}} \right\rbrack\end{matrix} \right. & (19)\end{matrix}$

When M number of packets are sent by N number of nodes, each node sends

$\frac{M}{N}$

number of packets due to the same energy utilization rate. The collidednumber of packets from all the nodes can be calculated as:

$\begin{matrix}{{\frac{M}{N}\left( {P_{2}^{(1)} + P_{3}^{(1)} + \ \ldots + P_{N}^{(1)}} \right)} + {\frac{M}{N}\left( {P_{2}^{(2)} + P_{3}^{(2)} + \ldots + P_{N}^{(2)}} \right)} + \ldots + {\frac{M}{N}\left( {P_{2}^{(N)} + P_{3}^{(N)} + \ldots + P_{N}^{(N)}} \right)}} & (20)\end{matrix}$

The collision probability for a network with N number of nodes is:

$\begin{matrix}{{\left( {{\frac{M}{N}\left( {P_{2}^{(1)} + P_{3}^{(1)} + \ldots + P_{N}^{(1)}} \right)} + {\frac{M}{N}\left( {P_{2}^{(2)} + P_{3}^{(2)} + \ldots + P_{N}^{(2)}} \right)} + \ldots + {\frac{M}{N}\left( {P_{2}^{(N)} + P_{3}^{(N)} + \ldots + P_{N}^{(N)}} \right)}} \right) \cdot \frac{1}{M}} = {\frac{1}{N}{\sum_{x = 1}^{N}{\sum_{y = 2}^{N}P_{y}^{(x)}}}}} & (21)\end{matrix}$

Note that the information transfer capacity for the whole network can becalculated using Eq. (4), Eq. (6), Eq. (7) and Eq. (21) according to thechoosing of time shift parameter Δ. The collision probability in Eq.(15), Eq. (16), Eq. (17), Eq. (18) and Eq. (21) involves the collisioncalculation from between two packets to between N packets, packetpositions go through all the states, and collision probability for allthe nodes. However, since the packets from a node cannot appear at allthe states, half of values in the probability matrix Eq. (14) are equalto zero. For example, the packets from Node-1 can never appear at states

$D_{2\Delta},D_{{2\Delta} + 1},\ldots \mspace{14mu},{D_{\frac{T{({N - 1})}}{\tau N} + {2\Delta} - 1}.}$

Therefore, the probability of the packets from Node-1 appearing at theabove states is equal to zero, namely,

$P_{D_{2\Delta}}^{(1)} = {P_{D_{{2\Delta} + 1}}^{(1)} = {P_{D_{\frac{T{({N - 1})}}{\tau N} + {2\Delta} - 1}}^{(1)} = {0.}}}$

These zero values in the probability matrix of Eq. (14) can reduce thecomputational complexity in Eq. (15), Eq. (16), Eq. (17) and Eq. (18).

For a Tx-only network application with a group of parameters (E, τ, W, Nand L), how to design time shift parameter Δ to maximize the informationtransfer capacity is important. First, the baseline inter-packetinterval is used to set the value of Δ with Δ=Δ_(T). Based on thepresent value Δ(Δ=Δ_(T)), the effective information capacity η_(Δ=Δ)_(T) can be calculated as follows using Eq. (4) and Eq. (21):

$\begin{matrix}\begin{matrix}{\eta_{\Delta = \Delta_{T}} = {\left( {1 - {\frac{1}{N}{\sum\limits_{x = 1}^{N}{\sum\limits_{y = 2}^{N}P_{y}^{(x)}}}}} \right) \cdot {\frac{L + \frac{\log_{2}\left( {\Delta!} \right)}{\Delta}}{\frac{L \cdot E}{W}}/\left( \frac{W}{E} \right)}}} \\{= {\frac{1}{N}{\sum_{x = 1}^{N}{\sum_{y = 2}^{N}{P_{y}^{(x)} \cdot \frac{L + \frac{\log_{2}\left( {\Delta!} \right)}{\Delta}}{L}}}}}}\end{matrix} & (22)\end{matrix}$

If the value η_(Δ=Δ) _(T) is less than one, this means that collisionhas significantly reduced the information capacity of SPPM. The optimalvalue of Δ falls in the range of

$\left\lbrack {\frac{T}{2\tau N},\frac{T}{\tau N}} \right\rbrack,$

which can be found through the calculation of effective informationcapacity η_(Δ) using Eq. (4) and Eq. (21).

If the value η_(Δ=Δ) _(T) is greater than one, the optimal value of Δeither falls in the range of

$\left\lbrack {\frac{T}{\tau N},\frac{T}{\tau}} \right\rbrack$

or is equal to

$\frac{2{L \cdot E}}{W \cdot \tau} - {2L} + {1.}$

After comparison of effective information capacity between these twopossible values, the optimal time shift parameter Δ can be assignedaccordingly.

The algorithm about searching for the optimal time shift parameter Δ tomaximum of the information transfer capacity using SPPM is given. Step1: calculate the effective information capacity η_(Δ=Δ) _(T) whenΔ=Δ_(T) using Eq. (22); If η_(Δ=Δ) _(T) =1, go to step 4. Otherwise, goto step 2 to calculate the effective information capacity η_(Δ) when

$\Delta \in \left\lbrack {\frac{T}{\tau N},\frac{T}{\tau}} \right\rbrack$

using Eq. (22), and find the maximum η_(Δ) ₁ and the corresponding timeshift value Δ₁. Then, go to step 3 and calculate the effectiveinformation capacity η_(Δ) ₂ when

$\Delta_{2} = {\frac{2{L \cdot E}}{W \cdot \tau} - {2L} + {1.}}$

Compare the two values of effective information capacity η_(Δ) ₁ andη_(Δ) ₂ . If η_(Δ) ₁ ≥η_(Δ) ₂ , then the optimal time shift Δ_(opt)=Δ₁.If

${\eta_{\Delta_{1}} < \eta_{\Delta_{2}}},{\Delta_{opt} = {\Delta_{2} = {\frac{2{L \cdot E}}{W \cdot \tau} - {2L} + {1.}}}}$

Next, go to step 5. At step 4, calculate the effective informationcapacity η_(Δ) when

$\Delta \in \left\lbrack {\frac{T}{2\tau \; N},\frac{T}{\tau N}} \right\rbrack$

using Eq. (22), and find the maximum η_(Δ) ₁ and the corresponding timeshift value Δ₁. The optimal time shift Δ_(opt)=Δ₁. Go to step 5. Step 5includes outputting the optimal time shift parameter Δ_(opt) and thecorresponding maximum effective information capacity η_(Δ) _(opt) .

Performance Analysis of SPPM

As shown above, a complete analysis has been developed to enhance theinformation capacity of sensor network without using extra energy. Theobtained information capacity using SPPM is always greater than thebaseline information capacity. The evaluation is implemented from twoperspectives about how the time shift parameter and number of nodesaffect the information transfer capacity. The effective informationcapacity concept is also used for the comparison purpose between SPPMinformation transfer capacity and the baseline one.

Note that energy utilization rate W=0.8 mW is used in the analysis,which corresponds to the baseline inter-packet interval is 1.6 seconds.However, the inter-packet interval is at least in the ranges of tens ofseconds in the real hardware implementation, which can moresignificantly benefit the performance of SPPM. A relatively smallinter-packet interval here is only for analysis purpose.

FIG. 14A shows the changes of effective information capacity with theincreasing of time shift Δ. FIG. 14B describes part of FIG. 14A with therange of

$\frac{\Delta}{\Delta_{T}} \in {\left\lbrack {0,{0.1}} \right\rbrack.}$

It can be seen in FIG. 14A that the maximum effective channel can beobtained at the same value of Δ, actually

$\Delta = {\frac{2{L \cdot E}}{W \cdot \tau} - {2L} + {1.}}$

Because when

${\frac{\Delta}{\Delta_{T}} \in \left\lbrack {1,{2 - \frac{{2L} - 1}{\Delta_{T}}}} \right\rbrack},$

the increasing of Δ within the range leads to the increasing of packetdistributed area (occupies more states), a wider area of packetdistribution does not incur the increasing of collision probabilitywhich can be verified in FIG. 15 that the collision probability for

$\frac{\Delta}{\Delta_{T}} \in \left\lbrack {1,{2 - \frac{{2L} - 1}{\Delta_{T}}}} \right\rbrack$

remains the same value. However, when

${\frac{\Delta}{\Delta_{T}} > {2 - \frac{{2L} - 1}{\Delta_{T}}}},$

even though the collision probability is decreasing due to the widerarea of packet position distribution, the average inter-packet intervalis increasing accordingly, which incurs the decreasing of effectiveinformation capacity. Therefore, the number of nodes in a network doesnot affect the value of optimal time shift Δ.

Whether the time shift Δ=2Δ_(T)−2L+1 is global optimal solution or localone, it depends on the collision probability. If a large collisionprobability worsens the effective information capacity as shown in FIG.15 with one hundred nodes, and the effective information capacity iseven less than 1, the value

$\Delta = {2 - \frac{{2L} - 1}{\Delta_{T}}}$

turns out be the local optimal solution. The global optimal solution canonly be achieved when Δ<Δ_(T). Therefore, it can be seen in FIG. 14Bthat the optimal value of Δ is less than 0.1Δ_(T). A relatively narrowpacket position distribution can mitigate the impact of collision. Forthe situation of the effective information capacity greater than 1 whenΔ=Δ_(T), a further investigation is needed for optimal Δ. For example,in FIGS. 14A and 14B, the optimal Δ for 30 nodes is within the range of[0, Δ_(T)], but the optimal Δ for 6 nodes is equal to 2Δ_(T)−2L+1, whichis greater than Δ_(T).

FIG. 16 shows how the maximum Effective Information Transfer Capacity(EITC) reduces with higher duty cycles for different network size. Thebaseline interval between two transmissions (i.e., T) determines theamount of free channel space available for the SPPM-WIS protocol toencode additional information by time-shifting the transmissions. Theoperating duty cycle is Lτ/T. Since T=L·E/W, duty cycle can be expressedas W·τ/E. In other words, for a given bit duration, and per-bittransmission energy budget E, duty cycle is determined by the energyharvesting rate W. Lower harvesting rates would cause lower duty cycles.

For each point in FIG. 16, the value of time shifts for which the EITCis maximized is marked. This is because with higher duty cycles, lesseramount of shifting space is available coupled with more frequentinter-node collision. Also, for larger networks, the maximum EITC issmaller due to more frequent collisions.

FIG. 17 shows how the increasing of number of nodes or network sizeaffects the effective information capacity and comparison between twoprotocols, CASPPM and SPPM. It can be seen in FIG. 17 that when thenumber of nodes in a network within a certain range (less than 40, forinstance), there is a clear difference between different time shiftparameters, the collision probability is also within a certain level(less than 20%) as shown in FIG. 18, the optimal time shift is alwaysequal to

$\Delta = {\frac{2{L \cdot E}}{W \cdot \tau} - {2L} + 1}$

to achieve the maximum effective information capacity. Therefore, Error!Reference source not found. shows that maximum theoretical effectiveinformation capacity 1 is highly overlapping with the curve for

$\Delta = {\frac{2{L \cdot E}}{W \cdot \tau} - {2L} + {1.}}$

Less number of nodes can vastly benefit the performance of SPPM, and thepresent level of collision is not big enough to lessen the throughput ofSPPM.

When the number of nodes increases to large enough (larger than 50),more than 20% of packets collide when Δ is greater than Δ_(T) as shownin FIG. 18. With the increasing of the number of nodes, the informationtransfer capacity of SPPM is even less than the baseline one (ECC<1).The optimal time shift Δ for maximum η can only be obtained in the rangeof [0, Δ_(T)], which tightens the packet position distribution within asmaller range and reduces the occurrence of collision. Therefore, theoptimal time shift Δ is less than Δ_(T) and it keeps decreasing with theincreasing of the number of nodes.

It can also be seen from FIG. 17 that the maximum η of SPPM is alwaysgreater than the maximum η of CASPPM. For a relatively small number ofnodes in a network, according to the analysis in Section IV.B.3), theoptimal Δ should be greater than Δ_(T). CASPPM protocol sets time shiftΔ less than Δ_(T), which does not make best use of inter-packet intervalto encode more data information. When a large number of nodes are in anetwork, CASPPM set a narrow range

$\left( \frac{T}{N} \right)$

to allow the packet for position modulation, which constrains themodulation of data. However, a slightly greater modulation space cancompensate the impact of collision from SPPM and achieves a greaterinformation capacity. Therefore, it can be seen in FIG. 17 that themaximum ECC of SPPM is always slightly greater than the maximum ECC ofCASPPM.

A lost packet has two distinct effects on the effective informationtransfer capacity. First, the raw information contained within thepacket is lost. Additionally, two separate pieces of the SPPM-codedinformation data is lost. One represented by the interval between thetransmission times of the last packet and the lost packet, and the otherrepresented by the interval between timings of the lost packet and thenext packet. FIG. 19 reports the maximum attainable effectiveinformation transfer capacity (EITC) of SPPM-WIS as a result ofdifferent packet loss probabilities due to channel errors. Theexperiments for these results are run such that for a given networksize, the shift value (i.e., Δ) that provides the maximum EITC ischosen.

As expected, the capacity diminishes monotonically with higher packetlosses, although the values remain larger than one for up to 10% packetlosses. Meaning SPPM-WIS still works better than the baseline case forup to 10% packet losses. It is notable that the decrease rate of EITCwith higher packet losses are very similar for all network sizes. Insummary, these results show that SPPM-WIS is deployable in networks withreasonable packet losses due to channel errors.

The techniques described herein or portions thereof may be implementedby one or more computer programs executed by one or more processors. Thecomputer programs include processor-executable instructions that arestored on a non-transitory tangible computer readable medium. Thecomputer programs may also include stored data. Non-limiting examples ofthe non-transitory tangible computer readable medium are nonvolatilememory, magnetic storage, and optical storage.

Some portions of the above description present the techniques describedherein in terms of algorithms and symbolic representations of operationson information. These algorithmic descriptions and representations arethe means used by those skilled in the data processing arts to mosteffectively convey the substance of their work to others skilled in theart. These operations, while described functionally or logically, areunderstood to be implemented by computer programs. Furthermore, it hasalso proven convenient at times to refer to these arrangements ofoperations as modules or by functional names, without loss ofgenerality.

Unless specifically stated otherwise as apparent from the abovediscussion, it is appreciated that throughout the description,discussions utilizing terms such as “processing” or “computing” or“calculating” or “determining” or “displaying” or the like, refer to theaction and processes of a computer system, or similar electroniccomputing device, that manipulates and transforms data represented asphysical (electronic) quantities within the computer system memories orregisters or other such information storage, transmission or displaydevices.

Certain aspects of the described techniques include process steps andinstructions described herein in the form of an algorithm. It should benoted that the described process steps and instructions could beembodied in software, firmware or hardware, and when embodied insoftware, could be downloaded to reside on and be operated fromdifferent platforms used by real time network operating systems.

The present disclosure also relates to an apparatus for performing theoperations herein. This apparatus may be specially constructed for therequired purposes, or it may comprise a computer selectively activatedor reconfigured by a computer program stored on a computer readablemedium that can be accessed by the computer. Such a computer program maybe stored in a tangible computer readable storage medium, such as, butis not limited to, any type of disk including floppy disks, opticaldisks, CD-ROMs, magnetic-optical disks, read-only memories (ROMs),random access memories (RAMs), EPROMs, EEPROMs, magnetic or opticalcards, application specific integrated circuits (ASICs), or any type ofmedia suitable for storing electronic instructions, and each coupled toa computer system bus. Furthermore, the computers referred to in thespecification may include a single processor or may be architecturesemploying multiple processor designs for increased computing capability.

The algorithms and operations presented herein are not inherentlyrelated to any particular computer or other apparatus. Various systemsmay also be used with programs in accordance with the teachings herein,or it may prove convenient to construct more specialized apparatuses toperform the required method steps. The required structure for a varietyof these systems will be apparent to those of skill in the art, alongwith equivalent variations. In addition, the present disclosure is notdescribed with reference to any particular programming language. It isappreciated that a variety of programming languages may be used toimplement the teachings of the present disclosure as described herein.

The foregoing description of the embodiments has been provided forpurposes of illustration and description. It is not intended to beexhaustive or to limit the disclosure. Individual elements or featuresof a particular embodiment are generally not limited to that particularembodiment, but, where applicable, are interchangeable and can be usedin a selected embodiment, even if not specifically shown or described.The same may also be varied in many ways. Such variations are not to beregarded as a departure from the disclosure, and all such modificationsare intended to be included within the scope of the disclosure.

What is claimed is:
 1. A packet position modulation system comprising: anode configured to transmit a plurality of packets at corresponding timeintervals, the node configured to adjust, for at least one packet of theplurality of packets, the corresponding time interval to transmit the atleast one packet; and a base station configured to: receive theplurality of packets from the node at corresponding time intervals;determine a difference between a previous time that a previous packet ofthe plurality of packets was received and a present time that a presentpacket of the plurality of packets was received; and recover coded datafrom the present packet based on the difference.
 2. The system of claim1 wherein the node includes a transmit time device configured to:receive sensor data from a sensor of the node; determine a delay basedon the sensor data; adjust the corresponding time interval based on thedetermined delay; and transmit the at least one packet in accordancewith the adjusted time interval.
 3. The system of claim 2 wherein thedifference indicates the sensor data.
 4. The system of claim 1 furthercomprising: an intermediary node including a transceiver and a sensor,the intermediary node configured to: receive the plurality of packetsfrom the node; and forward the plurality of packets to the base station.5. The system of claim 4 wherein the intermediary node includes anintermediary sensor configured to sense an environment condition at alocation of the intermediary sensor.
 6. The system of claim 1 whereinthe node is configured to: select a reference interval, and in responseto an energy level exceeding a threshold by: a present time equaling thereference interval less a next coded data, transmitting the presentpacket at the present time.
 7. The system of claim 6 wherein the node isconfigured to: in response to the energy level being below the thresholdby: the present time, transmitting the present packet at a postponedtime, wherein the postposed time equals the reference interval plus thenext coded data.
 8. The system of claim 7 wherein the next coded data isa next data multiplied by a packet duration.
 9. The system of claim 6wherein: the reference interval is selected as greater than or equal tothe corresponding time interval.
 10. The system of claim 1 wherein: thenode includes a sensor, and the plurality of packets include sensor datasensed by the sensor.
 11. The system of claim 1 wherein the base stationincludes a memory coupled to a processor, wherein the memory storesinstructions that, upon execution, cause the processor to recover thecoded data and store the previous time that the previous packet of theplurality of packets was received.
 12. The system of claim 1 wherein thebase station includes a display configured to display the present packetand the recovered coded data.
 13. A packet position modulation methodcomprising: transmitting, from a node, a plurality of packets atcorresponding time intervals; adjusting, by the node, for a first packetof the plurality of packets, a first time interval that the first packetis transmitted; receiving, at a base station, the first packet of theplurality of packets from the node at the first time interval;determining a difference between a previous time that a previous packetof the plurality of packets was received and a present time that thefirst packet of the plurality of packets was received; and recoveringcoded data from the first packet based on the difference.
 14. The methodof claim 13 further comprising: storing the present time that the firstpacket of the plurality of packets was received as the previous timethat the previous packet of the plurality of packets was received. 15.The method of claim 13 further comprising: receiving, at the node,sensor data from a sensor of the node; determining a time delay based onthe sensor data and a remaining energy level; adjusting the first timeinterval of the first packet based on the determined time delay; andtransmitting the first packet at the first time interval.
 16. The methodof claim 15 wherein the difference indicates the sensor data.
 17. Themethod of claim 13 further comprising: receiving, at an intermediarynode, the first packet from the node; and forwarding, from theintermediary node, the first packet to the base station.
 18. The methodof claim 17 wherein the intermediary node includes an intermediarysensor configured to sense an environment condition at a location of theintermediary sensor.
 19. The method of claim 18 further comprising:generating an intermediary packet including the first packet and theenvironment condition sensed at the location of the intermediary sensor,wherein the environment condition is included in the intermediary packetas the coded data; and forwarding, from the intermediary node, theintermediary packet to the base station.
 20. A packet positionmodulation system comprising: a node configured to transmit sensor dataat corresponding time intervals, wherein the node includes a sensor anda transmit time device, wherein the transmit time device is configuredto: receive sensor data from the sensor; determine a delay based on thesensor data and a remaining energy level; adjust the corresponding timeinterval based on the delay; and transmit the sensor data in accordancewith the adjusted time interval; and a base station configured toreceive sensor data from the node at the corresponding time interval,wherein the base station includes a memory coupled to a processor,wherein the memory stores instructions that, upon execution, cause theprocessor to: determine an actual period between receiving previoussensor data and the sensor data; and calculate the sensor data based onthe actual period.